Equation (REF ) is deduced by conditioning both terms in (REF ) with respect to \( {\cal F }_t\) , exchanging the (stochastic) integration with the conditional expectation for all the terms on the right hand side of (REF ). To justify this Fubini-like exchange one can use, for example, Lemma 3.21 in [1]}. One also needs to use the fact that, for arbitrary \(0\le s \le t\) , \(\widehat{m}_s=\mathbb {E}(m_s~|~ {\cal F }_s)=\mathbb {E}(m_s~|~ {\cal F }_t)\)  and \( \widehat{p}_s:=\mathbb {E}(p_s~|~ {\cal F }_s)=\mathbb {E}(p_s~|~ {\cal F }_t)\) .
To justify this, one can use, for example, Proposition 3.15 in [1]}. We note that both Proposition 3.15 and Lemma 3.21 hold true when the conditioning is done with respect the equivalent probability measure \(\tilde{\mathbb {P}}\)  under which \(Y\)  is a Brownian motion. However since
\((\overline{W}_{t},\overline{V}_{t},\overline{X}_{0})\)  are independent of \(Y\)  under \(\mathbb {P}\)  they remain independent of \(Y\)  under \(\tilde{\mathbb {P}}\)  and therefore conditioning the processes \(\xi ^i\)  under \(\tilde{\mathbb {P}}\)  is the same as
conditioning them under \(\mathbb {P}\) . Note that argument this does not apply to the original signal \(X\) .
